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受欢迎的三角函数 >

arcsin(3x)+arcsin(x)=60

解答

arcsin (3 x) + arcsin (x) = 6 0^{\circ}

解答

x = \frac{3}{2 13}

求解步骤

arcsin (3 x) + arcsin (x) = 6 0^{\circ}

使用三角恒等式改写

arcsin (3 x) + arcsin (x)

使用和差化积恒等式:

arcsin (s) + arcsin (t) = arcsin (s 1 - t^{2} + t 1 - s^{2})

= arcsin (3 x 1 - x^{2} + x 1 - (3 x)^{2})

arcsin (3 x 1 - x^{2} + x 1 - (3 x)^{2}) = 6 0^{\circ}

使用反三角函数性质

arcsin (3 x 1 - x^{2} + x 1 - (3 x)^{2}) = 6 0^{\circ}

arcsin (x) = a \Rightarrow x = sin (a)

3 x 1 - x^{2} + x 1 - (3 x)^{2} = sin (6 0^{\circ})

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

使用以下普通恒等式:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{3}{2}

3 x 1 - x^{2} + x 1 - (3 x)^{2} = \frac{3}{2}

3 x 1 - x^{2} + x 1 - (3 x)^{2} = \frac{3}{2}

解

3 x 1 - x^{2} + x 1 - (3 x)^{2} = \frac{3}{2} : x = \frac{3}{2 13}

3 x 1 - x^{2} + x 1 - (3 x)^{2} = \frac{3}{2}

在两边乘以

2

3 x 1 - x^{2} \cdot 2 + x 1 - (3 x)^{2} \cdot 2 = \frac{3}{2} \cdot 2

化简

6 1 - x^{2} x + 2 1 - (3 x)^{2} x = 3

去除平方根

6 1 - x^{2} x + 2 1 - (3 x)^{2} x = 3

两边减去

2 1 - (3 x)^{2} x

6 1 - x^{2} x + 2 1 - (3 x)^{2} x - 2 1 - (3 x)^{2} x = 3 - 2 1 - (3 x)^{2} x

化简

6 1 - x^{2} x = 3 - 2 1 - (3 x)^{2} x

两边进行平方:

36 x^{2} - 36 x^{4} = 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

6 1 - x^{2} x + 2 1 - (3 x)^{2} x = 3

(6 1 - x^{2} x)^{2} = (3 - 2 1 - (3 x)^{2} x)^{2}

展开

(6 1 - x^{2} x)^{2} : 36 x^{2} - 36 x^{4}

(6 1 - x^{2} x)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} x^{2} (1 - x^{2})^{2}

(1 - x^{2})^{2} : 1 - x^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((1 - x^{2})^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (1 - x^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 1 - x^{2}

= 6^{2} (1 - x^{2}) x^{2}

6^{2} = 36

= 36 (1 - x^{2}) x^{2}

展开

36 (1 - x^{2}) x^{2} : 36 x^{2} - 36 x^{4}

36 (1 - x^{2}) x^{2}

= 36 x^{2} (1 - x^{2})

使用分配律:

a (b - c) = ab - a c

a = 36 x^{2}, b = 1, c = x^{2}

= 36 x^{2} \cdot 1 - 36 x^{2} x^{2}

= 36 \cdot 1 \cdot x^{2} - 36 x^{2} x^{2}

化简

36 \cdot 1 \cdot x^{2} - 36 x^{2} x^{2} : 36 x^{2} - 36 x^{4}

36 \cdot 1 \cdot x^{2} - 36 x^{2} x^{2}

36 \cdot 1 \cdot x^{2} = 36 x^{2}

36 \cdot 1 \cdot x^{2}

数字相乘:

36 \cdot 1 = 36

= 36 x^{2}

36 x^{2} x^{2} = 36 x^{4}

36 x^{2} x^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 36 x^{2 + 2}

数字相加:

2 + 2 = 4

= 36 x^{4}

= 36 x^{2} - 36 x^{4}

= 36 x^{2} - 36 x^{4}

= 36 x^{2} - 36 x^{4}

展开

(3 - 2 1 - (3 x)^{2} x)^{2} : 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

(3 - 2 1 - (3 x)^{2} x)^{2}

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 3, b = 2 1 - (3 x)^{2} x

= (3)^{2} - 23 \cdot 2 1 - (3 x)^{2} x + (2 1 - (3 x)^{2} x)^{2}

化简

(3)^{2} - 23 \cdot 2 1 - (3 x)^{2} x + (2 1 - (3 x)^{2} x)^{2} : 3 - 43 1 - (3 x)^{2} x + 41 - (3 x)^{2} x^{2}

(3)^{2} - 23 \cdot 2 1 - (3 x)^{2} x + (2 1 - (3 x)^{2} x)^{2}

(3)^{2} = 3

(3)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

23 \cdot 2 1 - (3 x)^{2} x = 43 1 - (3 x)^{2} x

23 \cdot 2 1 - (3 x)^{2} x

数字相乘:

2 \cdot 2 = 4

= 43 1 - (3 x)^{2} x

(2 1 - (3 x)^{2} x)^{2} = 41 - (3 x)^{2} x^{2}

(2 1 - (3 x)^{2} x)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} x^{2} (1 - (3 x)^{2})^{2}

(1 - (3 x)^{2})^{2} : 1 - (3 x)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((1 - (3 x)^{2})^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (1 - (3 x)^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 1 - (3 x)^{2}

= 2^{2} (1 - (3 x)^{2}) x^{2}

2^{2} = 4

= 4 (1 - (3 x)^{2}) x^{2}

= 3 - 43 1 - (3 x)^{2} x + 4 (1 - (3 x)^{2}) x^{2}

= 3 - 43 1 - (3 x)^{2} x + 4 (1 - (3 x)^{2}) x^{2}

展开

3 - 43 1 - (3 x)^{2} x + 4 (1 - (3 x)^{2}) x^{2} : 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

3 - 43 1 - (3 x)^{2} x + 4 (1 - (3 x)^{2}) x^{2}

1 - (3 x)^{2} = 1 - 9 x^{2}

1 - (3 x)^{2}

(3 x)^{2} = 9 x^{2}

(3 x)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 3^{2} x^{2}

3^{2} = 9

= 9 x^{2}

= 1 - 9 x^{2}

= 3 - 43 x - 9 x^{2} + 1 + 4 x^{2} (- (3 x)^{2} + 1)

(3 x)^{2} = 9 x^{2}

(3 x)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 3^{2} x^{2}

3^{2} = 9

= 9 x^{2}

= 3 - 43 x - 9 x^{2} + 1 + 4 x^{2} (- 9 x^{2} + 1)

= 3 - 43 x 1 - 9 x^{2} + 4 x^{2} (1 - 9 x^{2})

乘开

4 x^{2} (1 - 9 x^{2}) : 4 x^{2} - 36 x^{4}

4 x^{2} (1 - 9 x^{2})

使用分配律:

a (b - c) = ab - a c

a = 4 x^{2}, b = 1, c = 9 x^{2}

= 4 x^{2} \cdot 1 - 4 x^{2} \cdot 9 x^{2}

= 4 \cdot 1 \cdot x^{2} - 4 \cdot 9 x^{2} x^{2}

化简

4 \cdot 1 \cdot x^{2} - 4 \cdot 9 x^{2} x^{2} : 4 x^{2} - 36 x^{4}

4 \cdot 1 \cdot x^{2} - 4 \cdot 9 x^{2} x^{2}

4 \cdot 1 \cdot x^{2} = 4 x^{2}

4 \cdot 1 \cdot x^{2}

数字相乘:

4 \cdot 1 = 4

= 4 x^{2}

4 \cdot 9 x^{2} x^{2} = 36 x^{4}

4 \cdot 9 x^{2} x^{2}

数字相乘:

4 \cdot 9 = 36

= 36 x^{2} x^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 36 x^{2 + 2}

数字相加:

2 + 2 = 4

= 36 x^{4}

= 4 x^{2} - 36 x^{4}

= 4 x^{2} - 36 x^{4}

= 3 - 43 1 - 9 x^{2} x + 4 x^{2} - 36 x^{4}

= 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

= 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

36 x^{2} - 36 x^{4} = 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

36 x^{2} - 36 x^{4} = 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

两边减去

4 x^{2} - 36 x^{4}

36 x^{2} - 36 x^{4} - (4 x^{2} - 36 x^{4}) = 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4} - (4 x^{2} - 36 x^{4})

化简

32 x^{2} = - 43 1 - 9 x^{2} x + 3

两边减去

3

32 x^{2} - 3 = - 43 1 - 9 x^{2} x + 3 - 3

化简

32 x^{2} - 3 = - 43 1 - 9 x^{2} x

两边进行平方:

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

36 x^{2} - 36 x^{4} = 3 - 43 x 1 - 9 x^{2} + 4 x^{2} - 36 x^{4}

(32 x^{2} - 3)^{2} = (- 43 1 - 9 x^{2} x)^{2}

展开

(32 x^{2} - 3)^{2} : 1024 x^{4} - 192 x^{2} + 9

(32 x^{2} - 3)^{2}

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 32 x^{2}, b = 3

= (32 x^{2})^{2} - 2 \cdot 32 x^{2} \cdot 3 + 3^{2}

化简

(32 x^{2})^{2} - 2 \cdot 32 x^{2} \cdot 3 + 3^{2} : 1024 x^{4} - 192 x^{2} + 9

(32 x^{2})^{2} - 2 \cdot 32 x^{2} \cdot 3 + 3^{2}

(32 x^{2})^{2} = 1024 x^{4}

(32 x^{2})^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 3 2^{2} (x^{2})^{2}

(x^{2})^{2} : x^{4}

使用指数法则:

(a^{b})^{c} = a^{b c}

= x^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= x^{4}

= 3 2^{2} x^{4}

3 2^{2} = 1024

= 1024 x^{4}

2 \cdot 32 x^{2} \cdot 3 = 192 x^{2}

2 \cdot 32 x^{2} \cdot 3

数字相乘:

2 \cdot 32 \cdot 3 = 192

= 192 x^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

= 1024 x^{4} - 192 x^{2} + 9

= 1024 x^{4} - 192 x^{2} + 9

展开

(- 43 1 - 9 x^{2} x)^{2} : 48 x^{2} - 432 x^{4}

(- 43 1 - 9 x^{2} x)^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 43 1 - 9 x^{2} x)^{2} = (43 1 - 9 x^{2} x)^{2}

= (43 1 - 9 x^{2} x)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (3)^{2} x^{2} (1 - 9 x^{2})^{2}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= 4^{2} \cdot 3 (1 - 9 x^{2})^{2} x^{2}

(1 - 9 x^{2})^{2} : 1 - 9 x^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((1 - 9 x^{2})^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (1 - 9 x^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 1 - 9 x^{2}

= 4^{2} \cdot 3 (1 - 9 x^{2}) x^{2}

整理后得

= 48 (1 - 9 x^{2}) x^{2}

展开

48 (1 - 9 x^{2}) x^{2} : 48 x^{2} - 432 x^{4}

48 (1 - 9 x^{2}) x^{2}

= 48 x^{2} (1 - 9 x^{2})

使用分配律:

a (b - c) = ab - a c

a = 48 x^{2}, b = 1, c = 9 x^{2}

= 48 x^{2} \cdot 1 - 48 x^{2} \cdot 9 x^{2}

= 48 \cdot 1 \cdot x^{2} - 48 \cdot 9 x^{2} x^{2}

化简

48 \cdot 1 \cdot x^{2} - 48 \cdot 9 x^{2} x^{2} : 48 x^{2} - 432 x^{4}

48 \cdot 1 \cdot x^{2} - 48 \cdot 9 x^{2} x^{2}

48 \cdot 1 \cdot x^{2} = 48 x^{2}

48 \cdot 1 \cdot x^{2}

数字相乘:

48 \cdot 1 = 48

= 48 x^{2}

48 \cdot 9 x^{2} x^{2} = 432 x^{4}

48 \cdot 9 x^{2} x^{2}

数字相乘:

48 \cdot 9 = 432

= 432 x^{2} x^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 432 x^{2 + 2}

数字相加:

2 + 2 = 4

= 432 x^{4}

= 48 x^{2} - 432 x^{4}

= 48 x^{2} - 432 x^{4}

= 48 x^{2} - 432 x^{4}

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

解

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4} : x = \frac{3}{2 7}, x = - \frac{3}{2 7}, x = \frac{3}{2 13}, x = - \frac{3}{2 13}

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

将

432 x^{4}

para o lado esquerdo

1024 x^{4} - 192 x^{2} + 9 = 48 x^{2} - 432 x^{4}

两边加上

432 x^{4}

1024 x^{4} - 192 x^{2} + 9 + 432 x^{4} = 48 x^{2} - 432 x^{4} + 432 x^{4}

化简

1456 x^{4} - 192 x^{2} + 9 = 48 x^{2}

1456 x^{4} - 192 x^{2} + 9 = 48 x^{2}

将

48 x^{2}

para o lado esquerdo

1456 x^{4} - 192 x^{2} + 9 = 48 x^{2}

两边减去

48 x^{2}

1456 x^{4} - 192 x^{2} + 9 - 48 x^{2} = 48 x^{2} - 48 x^{2}

化简

1456 x^{4} - 240 x^{2} + 9 = 0

1456 x^{4} - 240 x^{2} + 9 = 0

两边除以

1456

\frac{1456 x ^{4}}{1456} - \frac{240 x ^{2}}{1456} + \frac{9}{1456} = \frac{0}{1456}

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

x^{4} - \frac{15 x ^{2}}{91} + \frac{9}{1456} = 0

用

u = x^{2}

和

u^{2} = x^{4}

改写方程式

u^{2} - \frac{15 u}{91} + \frac{9}{1456} = 0

解

u^{2} - \frac{15 u}{91} + \frac{9}{1456} = 0 : u = \frac{3}{28}, u = \frac{3}{52}

u^{2} - \frac{15 u}{91} + \frac{9}{1456} = 0

找到

91, 1456

的最小公倍数:

1456

91, 1456

最小公倍数 (LCM)

91

质因数分解:

7 \cdot 13

91

91

除以

791 = 13 \cdot 7

= 7 \cdot 13

7, 13

都是质数，因此无法进一步因数分解

= 7 \cdot 13

1456

质因数分解:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 7 \cdot 13

1456

1456

除以

21456 = 728 \cdot 2

= 2 \cdot 728

728

除以

2728 = 364 \cdot 2

= 2 \cdot 2 \cdot 364

364

除以

2364 = 182 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 182

182

除以

2182 = 91 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 91

91

除以

791 = 13 \cdot 7

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7 \cdot 13

2, 7, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7 \cdot 13

将每个因子乘以它在

91

或

1456

中出现的最多次数

= 7 \cdot 13 \cdot 2 \cdot 2 \cdot 2 \cdot 2

数字相乘:

7 \cdot 13 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 1456

= 1456

乘以最小公倍数=

1456

u^{2} \cdot 1456 - \frac{15 u}{91} \cdot 1456 + \frac{9}{1456} \cdot 1456 = 0 \cdot 1456

化简

1456 u^{2} - 240 u + 9 = 0

两边除以

1456

\frac{1456 u ^{2}}{1456} - \frac{240 u}{1456} + \frac{9}{1456} = \frac{0}{1456}

改写成标准形式

a x^{2} + b x + c = 0

u^{2} - \frac{15 u}{91} + \frac{9}{1456} = 0

使用求根公式求解

u^{2} - \frac{15 u}{91} + \frac{9}{1456} = 0

二次方程求根公式:

若

a = 1, b = - \frac{15}{91}, c = \frac{9}{1456}

u_{1, 2} = \frac{- ( - \frac{15}{91} ) \pm ( - \frac{15}{91} ) ^{2} - 4 \cdot 1 \cdot \frac{9}{1456}}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{15}{91} ) \pm ( - \frac{15}{91} ) ^{2} - 4 \cdot 1 \cdot \frac{9}{1456}}{2 \cdot 1}

(- \frac{15}{91})^{2} - 4 \cdot 1 \cdot \frac{9}{1456} = \frac{9}{182}

(- \frac{15}{91})^{2} - 4 \cdot 1 \cdot \frac{9}{1456}

(- \frac{15}{91})^{2} = \frac{1 5 ^{2}}{9 1 ^{2}}

(- \frac{15}{91})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{15}{91})^{2} = (\frac{15}{91})^{2}

= (\frac{15}{91})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 5 ^{2}}{9 1 ^{2}}

4 \cdot 1 \cdot \frac{9}{1456} = \frac{9}{364}

4 \cdot 1 \cdot \frac{9}{1456}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{9 \cdot 4}{1456}

\frac{9 \cdot 4}{1456} = \frac{9}{364}

\frac{9 \cdot 4}{1456}

数字相乘:

9 \cdot 4 = 36

= \frac{36}{1456}

约分:

4

= \frac{9}{364}

= 1 \cdot \frac{9}{364}

乘以:

1 \cdot \frac{9}{364} = \frac{9}{364}

= \frac{9}{364}

= \frac{1 5 ^{2}}{9 1 ^{2}} - \frac{9}{364}

\frac{1 5 ^{2}}{9 1 ^{2}} = \frac{225}{8281}

\frac{1 5 ^{2}}{9 1 ^{2}}

1 5^{2} = 225

= \frac{225}{9 1 ^{2}}

9 1^{2} = 8281

= \frac{225}{8281}

= \frac{225}{8281} - \frac{9}{364}

化简

\frac{225}{8281} - \frac{9}{364} : \frac{81}{33124}

\frac{225}{8281} - \frac{9}{364}

8281, 364

的最小公倍数:

33124

8281, 364

最小公倍数 (LCM)

8281

质因数分解:

7 \cdot 7 \cdot 13 \cdot 13

8281

8281

除以

78281 = 1183 \cdot 7

= 7 \cdot 1183

1183

除以

71183 = 169 \cdot 7

= 7 \cdot 7 \cdot 169

169

除以

13169 = 13 \cdot 13

= 7 \cdot 7 \cdot 13 \cdot 13

7, 13

都是质数，因此无法进一步因数分解

= 7 \cdot 7 \cdot 13 \cdot 13

364

质因数分解:

2 \cdot 2 \cdot 7 \cdot 13

364

364

除以

2364 = 182 \cdot 2

= 2 \cdot 182

182

除以

2182 = 91 \cdot 2

= 2 \cdot 2 \cdot 91

91

除以

791 = 13 \cdot 7

= 2 \cdot 2 \cdot 7 \cdot 13

2, 7, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7 \cdot 13

将每个因子乘以它在

8281

或

364

中出现的最多次数

= 7 \cdot 7 \cdot 13 \cdot 13 \cdot 2 \cdot 2

数字相乘:

7 \cdot 7 \cdot 13 \cdot 13 \cdot 2 \cdot 2 = 33124

= 33124

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

33124

对于

\frac{225}{8281} :

将分母和分子乘以

4

\frac{225}{8281} = \frac{225 \cdot 4}{8281 \cdot 4} = \frac{900}{33124}

对于

\frac{9}{364} :

将分母和分子乘以

91

\frac{9}{364} = \frac{9 \cdot 91}{364 \cdot 91} = \frac{819}{33124}

= \frac{900}{33124} - \frac{819}{33124}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{900 - 819}{33124}

数字相减:

900 - 819 = 81

= \frac{81}{33124}

= \frac{81}{33124}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{81}{33124}

33124 = 182

33124

因式分解数字:

33124 = 18 2^{2}

= 18 2^{2}

使用根式运算法则:

n a^{n} = a

18 2^{2} = 182

= 182

= \frac{81}{182}

81 = 9

81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

= \frac{9}{182}

u_{1, 2} = \frac{- ( - \frac{15}{91} ) \pm \frac{9}{182}}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - \frac{15}{91} ) + \frac{9}{182}}{2 \cdot 1}, u_{2} = \frac{- ( - \frac{15}{91} ) - \frac{9}{182}}{2 \cdot 1}

u = \frac{- ( - \frac{15}{91} ) + \frac{9}{182}}{2 \cdot 1} : \frac{3}{28}

\frac{- ( - \frac{15}{91} ) + \frac{9}{182}}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{\frac{15}{91} + \frac{9}{182}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{15}{91} + \frac{9}{182}}{2}

化简

\frac{15}{91} + \frac{9}{182} : \frac{3}{14}

\frac{15}{91} + \frac{9}{182}

91, 182

的最小公倍数:

182

91, 182

最小公倍数 (LCM)

91

质因数分解:

7 \cdot 13

91

91

除以

791 = 13 \cdot 7

= 7 \cdot 13

7, 13

都是质数，因此无法进一步因数分解

= 7 \cdot 13

182

质因数分解:

2 \cdot 7 \cdot 13

182

182

除以

2182 = 91 \cdot 2

= 2 \cdot 91

91

除以

791 = 13 \cdot 7

= 2 \cdot 7 \cdot 13

2, 7, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 7 \cdot 13

将每个因子乘以它在

91

或

182

中出现的最多次数

= 7 \cdot 13 \cdot 2

数字相乘:

7 \cdot 13 \cdot 2 = 182

= 182

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

182

对于

\frac{15}{91} :

将分母和分子乘以

2

\frac{15}{91} = \frac{15 \cdot 2}{91 \cdot 2} = \frac{30}{182}

= \frac{30}{182} + \frac{9}{182}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{30 + 9}{182}

数字相加:

30 + 9 = 39

= \frac{39}{182}

约分:

13

= \frac{3}{14}

= \frac{\frac{3}{14}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3}{14 \cdot 2}

数字相乘:

14 \cdot 2 = 28

= \frac{3}{28}

u = \frac{- ( - \frac{15}{91} ) - \frac{9}{182}}{2 \cdot 1} : \frac{3}{52}

\frac{- ( - \frac{15}{91} ) - \frac{9}{182}}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{\frac{15}{91} - \frac{9}{182}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{15}{91} - \frac{9}{182}}{2}

化简

\frac{15}{91} - \frac{9}{182} : \frac{3}{26}

\frac{15}{91} - \frac{9}{182}

91, 182

的最小公倍数:

182

91, 182

最小公倍数 (LCM)

91

质因数分解:

7 \cdot 13

91

91

除以

791 = 13 \cdot 7

= 7 \cdot 13

7, 13

都是质数，因此无法进一步因数分解

= 7 \cdot 13

182

质因数分解:

2 \cdot 7 \cdot 13

182

182

除以

2182 = 91 \cdot 2

= 2 \cdot 91

91

除以

791 = 13 \cdot 7

= 2 \cdot 7 \cdot 13

2, 7, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 7 \cdot 13

将每个因子乘以它在

91

或

182

中出现的最多次数

= 7 \cdot 13 \cdot 2

数字相乘:

7 \cdot 13 \cdot 2 = 182

= 182

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

182

对于

\frac{15}{91} :

将分母和分子乘以

2

\frac{15}{91} = \frac{15 \cdot 2}{91 \cdot 2} = \frac{30}{182}

= \frac{30}{182} - \frac{9}{182}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{30 - 9}{182}

数字相减:

30 - 9 = 21

= \frac{21}{182}

约分:

7

= \frac{3}{26}

= \frac{\frac{3}{26}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3}{26 \cdot 2}

数字相乘:

26 \cdot 2 = 52

= \frac{3}{52}

二次方程组的解是:

u = \frac{3}{28}, u = \frac{3}{52}

u = \frac{3}{28}, u = \frac{3}{52}

代回

u = x^{2}

，求解

x

解

x^{2} = \frac{3}{28} : x = \frac{3}{2 7}, x = - \frac{3}{2 7}

x^{2} = \frac{3}{28}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

x = \frac{3}{28}, x = - \frac{3}{28}

\frac{3}{28} = \frac{3}{2 7}

\frac{3}{28}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{3}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 7 = 2^{2} 7

= 2^{2} 7

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 27

= \frac{3}{2 7}

- \frac{3}{28} = - \frac{3}{2 7}

- \frac{3}{28}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{3}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 7 = 2^{2} 7

= 2^{2} 7

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 27

= - \frac{3}{2 7}

x = \frac{3}{2 7}, x = - \frac{3}{2 7}

解

x^{2} = \frac{3}{52} : x = \frac{3}{2 13}, x = - \frac{3}{2 13}

x^{2} = \frac{3}{52}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

x = \frac{3}{52}, x = - \frac{3}{52}

\frac{3}{52} = \frac{3}{2 13}

\frac{3}{52}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{3}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 13 = 2^{2} 13

= 2^{2} 13

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 213

= \frac{3}{2 13}

- \frac{3}{52} = - \frac{3}{2 13}

- \frac{3}{52}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{3}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

ab = a b, a \geq 0, b \geq 0

2^{2} \cdot 13 = 2^{2} 13

= 2^{2} 13

使用根式运算法则:

a^{2} = a, a \geq 0

2^{2} = 2

= 213

= - \frac{3}{2 13}

x = \frac{3}{2 13}, x = - \frac{3}{2 13}

解为

x = \frac{3}{2 7}, x = - \frac{3}{2 7}, x = \frac{3}{2 13}, x = - \frac{3}{2 13}

x = \frac{3}{2 7}, x = - \frac{3}{2 7}, x = \frac{3}{2 13}, x = - \frac{3}{2 13}

验证解:

x = \frac{3}{2 7}

假

, x = - \frac{3}{2 7}

假

, x = \frac{3}{2 13}

真

, x = - \frac{3}{2 13}

假

将它们代入

3 x 1 - x^{2} + x 1 - (3 x)^{2} = \frac{3}{2}

检验解是否符合

去除与方程不符的解。

代入

x = \frac{3}{2 7} :

假

3 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2} + (\frac{3}{2 7}) 1 - (3 (\frac{3}{2 7}))^{2} = \frac{3}{2}

3 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2} + (\frac{3}{2 7}) 1 - (3 (\frac{3}{2 7}))^{2} = \frac{4 3}{7}

3 (\frac{3}{2 7}) 1 - (\frac{3}{2 7})^{2} + (\frac{3}{2 7}) 1 - (3 (\frac{3}{2 7}))^{2}

去除括号:

(a) = a

= 3 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2} + \frac{3}{2 7} 1 - (3 \cdot \frac{3}{2 7})^{2}

3 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2} = \frac{15 3}{28}

3 \cdot \frac{3}{2 7} 1 - (\frac{3}{2 7})^{2}

1 - (\frac{3}{2 7})^{2} = \frac{5}{2 7}

1 - (\frac{3}{2 7})^{2}

(\frac{3}{2 7})^{2} = \frac{3}{28}

(\frac{3}{2 7})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 7

= \frac{3}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= 1 - \frac{3}{28}

化简

1 - \frac{3}{28} : \frac{25}{28}

1 - \frac{3}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{3}{28}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 28 - 3}{28}

1 \cdot 28 - 3 = 25

1 \cdot 28 - 3

数字相乘:

1 \cdot 28 = 28

= 28 - 3

数字相减:

28 - 3 = 25

= 25

= \frac{25}{28}

= \frac{25}{28}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{25}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

= \frac{25}{2 7}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

= \frac{5}{2 7}

= 3 \cdot \frac{5}{2 7} \cdot \frac{3}{2 7}

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 5 \cdot 3}{2 7 \cdot 2 7}

数字相乘:

5 \cdot 3 = 15

= \frac{15 3}{2 \cdot 2 7 7}

27 \cdot 27 = 28

27 \cdot 27

数字相乘:

2 \cdot 2 = 4

= 477

使用根式运算法则:

a a = a

77 = 7

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{15 3}{28}

\frac{3}{2 7} 1 - (3 \cdot \frac{3}{2 7})^{2} = \frac{3}{28}

\frac{3}{2 7} 1 - (3 \cdot \frac{3}{2 7})^{2}

1 - (3 \cdot \frac{3}{2 7})^{2} = \frac{1}{2 7}

1 - (3 \cdot \frac{3}{2 7})^{2}

(3 \cdot \frac{3}{2 7})^{2} = \frac{27}{28}

(3 \cdot \frac{3}{2 7})^{2}

乘

3 \cdot \frac{3}{2 7} : \frac{3 3}{2 7}

3 \cdot \frac{3}{2 7}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 3}{2 7}

= (\frac{3 \cdot 3}{2 7})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(33)^{2} = 3^{2} (3)^{2}

= \frac{3 ^{2} ( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 7

= \frac{( 3 ) ^{2} \cdot 3 ^{2}}{2 ^{2} \cdot 7}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3 \cdot 3 ^{2}}{2 ^{2} \cdot 7}

3 \cdot 3^{2} = 3^{3}

3 \cdot 3^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

3 \cdot 3^{2} = 3^{1 + 2}

= 3^{1 + 2}

数字相加:

1 + 2 = 3

= 3^{3}

= \frac{3 ^{3}}{2 ^{2} \cdot 7}

3^{3} = 27

= \frac{27}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{27}{28}

= 1 - \frac{27}{28}

化简

1 - \frac{27}{28} : \frac{1}{28}

1 - \frac{27}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{27}{28}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 28 - 27}{28}

1 \cdot 28 - 27 = 1

1 \cdot 28 - 27

数字相乘:

1 \cdot 28 = 28

= 28 - 27

数字相减:

28 - 27 = 1

= 1

= \frac{1}{28}

= \frac{1}{28}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

= \frac{1}{2 7}

使用法则

1 = 1

= \frac{1}{2 7}

= \frac{1}{2 7} \cdot \frac{3}{2 7}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{3 \cdot 1}{2 7 \cdot 2 7}

乘以:

3 \cdot 1 = 3

= \frac{3}{2 \cdot 2 7 7}

27 \cdot 27 = 28

27 \cdot 27

数字相乘:

2 \cdot 2 = 4

= 477

使用根式运算法则:

a a = a

77 = 7

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= \frac{15 3}{28} + \frac{3}{28}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{15 3 + 3}{28}

同类项相加:

153 + 3 = 163

= \frac{16 3}{28}

约分:

4

= \frac{4 3}{7}

\frac{4 3}{7} = \frac{3}{2}

假

代入

x = - \frac{3}{2 7} :

假

3 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2} + (- \frac{3}{2 7}) 1 - (3 (- \frac{3}{2 7}))^{2} = \frac{3}{2}

3 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2} + (- \frac{3}{2 7}) 1 - (3 (- \frac{3}{2 7}))^{2} = - \frac{4 3}{7}

3 (- \frac{3}{2 7}) 1 - (- \frac{3}{2 7})^{2} + (- \frac{3}{2 7}) 1 - (3 (- \frac{3}{2 7}))^{2}

去除括号:

(- a) = - a

= - 3 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2} - \frac{3}{2 7} 1 - (- 3 \cdot \frac{3}{2 7})^{2}

3 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2} = \frac{15 3}{28}

3 \cdot \frac{3}{2 7} 1 - (- \frac{3}{2 7})^{2}

1 - (- \frac{3}{2 7})^{2} = \frac{5}{2 7}

1 - (- \frac{3}{2 7})^{2}

(- \frac{3}{2 7})^{2} = \frac{3}{28}

(- \frac{3}{2 7})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 7})^{2} = (\frac{3}{2 7})^{2}

= (\frac{3}{2 7})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 7

= \frac{3}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= 1 - \frac{3}{28}

化简

1 - \frac{3}{28} : \frac{25}{28}

1 - \frac{3}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{3}{28}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 28 - 3}{28}

1 \cdot 28 - 3 = 25

1 \cdot 28 - 3

数字相乘:

1 \cdot 28 = 28

= 28 - 3

数字相减:

28 - 3 = 25

= 25

= \frac{25}{28}

= \frac{25}{28}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{25}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

= \frac{25}{2 7}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

= \frac{5}{2 7}

= 3 \cdot \frac{5}{2 7} \cdot \frac{3}{2 7}

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 5 \cdot 3}{2 7 \cdot 2 7}

数字相乘:

5 \cdot 3 = 15

= \frac{15 3}{2 \cdot 2 7 7}

27 \cdot 27 = 28

27 \cdot 27

数字相乘:

2 \cdot 2 = 4

= 477

使用根式运算法则:

a a = a

77 = 7

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{15 3}{28}

\frac{3}{2 7} 1 - (- 3 \cdot \frac{3}{2 7})^{2} = \frac{3}{28}

\frac{3}{2 7} 1 - (- 3 \cdot \frac{3}{2 7})^{2}

1 - (- 3 \cdot \frac{3}{2 7})^{2} = \frac{1}{2 7}

1 - (- 3 \cdot \frac{3}{2 7})^{2}

(- 3 \cdot \frac{3}{2 7})^{2} = \frac{27}{28}

(- 3 \cdot \frac{3}{2 7})^{2}

乘

- 3 \cdot \frac{3}{2 7} : - \frac{3 3}{2 7}

- 3 \cdot \frac{3}{2 7}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{3 \cdot 3}{2 7}

= (- \frac{3 3}{2 7})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3 3}{2 7})^{2} = (\frac{3 \cdot 3}{2 7})^{2}

= (\frac{3 \cdot 3}{2 7})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 3 ) ^{2}}{( 2 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(27)^{2} = 2^{2} (7)^{2}

= \frac{( 3 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(33)^{2} = 3^{2} (3)^{2}

= \frac{3 ^{2} ( 3 ) ^{2}}{2 ^{2} ( 7 ) ^{2}}

(7)^{2} : 7

使用根式运算法则:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 7

= \frac{( 3 ) ^{2} \cdot 3 ^{2}}{2 ^{2} \cdot 7}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3 \cdot 3 ^{2}}{2 ^{2} \cdot 7}

3 \cdot 3^{2} = 3^{3}

3 \cdot 3^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

3 \cdot 3^{2} = 3^{1 + 2}

= 3^{1 + 2}

数字相加:

1 + 2 = 3

= 3^{3}

= \frac{3 ^{3}}{2 ^{2} \cdot 7}

3^{3} = 27

= \frac{27}{2 ^{2} \cdot 7}

2^{2} \cdot 7 = 28

2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{27}{28}

= 1 - \frac{27}{28}

化简

1 - \frac{27}{28} : \frac{1}{28}

1 - \frac{27}{28}

将项转换为分式:

1 = \frac{1 \cdot 28}{28}

= \frac{1 \cdot 28}{28} - \frac{27}{28}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 28 - 27}{28}

1 \cdot 28 - 27 = 1

1 \cdot 28 - 27

数字相乘:

1 \cdot 28 = 28

= 28 - 27

数字相减:

28 - 27 = 1

= 1

= \frac{1}{28}

= \frac{1}{28}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{28}

28 = 27

28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

= \frac{1}{2 7}

使用法则

1 = 1

= \frac{1}{2 7}

= \frac{1}{2 7} \cdot \frac{3}{2 7}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{3 \cdot 1}{2 7 \cdot 2 7}

乘以:

3 \cdot 1 = 3

= \frac{3}{2 \cdot 2 7 7}

27 \cdot 27 = 28

27 \cdot 27

数字相乘:

2 \cdot 2 = 4

= 477

使用根式运算法则:

a a = a

77 = 7

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= \frac{3}{28}

= - \frac{15 3}{28} - \frac{3}{28}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 15 3 - 3}{28}

同类项相加:

- 153 - 3 = - 163

= \frac{- 16 3}{28}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{16 3}{28}

约分:

4

= - \frac{4 3}{7}

- \frac{4 3}{7} = \frac{3}{2}

假

代入

x = \frac{3}{2 13} :

真

3 (\frac{3}{2 13}) 1 - (\frac{3}{2 13})^{2} + (\frac{3}{2 13}) 1 - (3 (\frac{3}{2 13}))^{2} = \frac{3}{2}

3 (\frac{3}{2 13}) 1 - (\frac{3}{2 13})^{2} + (\frac{3}{2 13}) 1 - (3 (\frac{3}{2 13}))^{2} = \frac{3}{2}

3 (\frac{3}{2 13}) 1 - (\frac{3}{2 13})^{2} + (\frac{3}{2 13}) 1 - (3 (\frac{3}{2 13}))^{2}

去除括号:

(a) = a

= 3 \cdot \frac{3}{2 13} 1 - (\frac{3}{2 13})^{2} + \frac{3}{2 13} 1 - (3 \cdot \frac{3}{2 13})^{2}

3 \cdot \frac{3}{2 13} 1 - (\frac{3}{2 13})^{2} = \frac{21 3}{52}

3 \cdot \frac{3}{2 13} 1 - (\frac{3}{2 13})^{2}

1 - (\frac{3}{2 13})^{2} = \frac{7}{2 13}

1 - (\frac{3}{2 13})^{2}

(\frac{3}{2 13})^{2} = \frac{3}{52}

(\frac{3}{2 13})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(213)^{2} = 2^{2} (13)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 13 ) ^{2}}

(13)^{2} : 13

使用根式运算法则:

a = a^{\frac{1}{2}}

= (1 3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 1 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 13

= \frac{3}{2 ^{2} \cdot 13}

2^{2} \cdot 13 = 52

2^{2} \cdot 13

2^{2} = 4

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{3}{52}

= 1 - \frac{3}{52}

化简

1 - \frac{3}{52} : \frac{49}{52}

1 - \frac{3}{52}

将项转换为分式:

1 = \frac{1 \cdot 52}{52}

= \frac{1 \cdot 52}{52} - \frac{3}{52}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 52 - 3}{52}

1 \cdot 52 - 3 = 49

1 \cdot 52 - 3

数字相乘:

1 \cdot 52 = 52

= 52 - 3

数字相减:

52 - 3 = 49

= 49

= \frac{49}{52}

= \frac{49}{52}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{49}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

n ab = n a n b

= 13 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 213

= \frac{49}{2 13}

49 = 7

49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

= \frac{7}{2 13}

= 3 \cdot \frac{7}{2 13} \cdot \frac{3}{2 13}

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 7 \cdot 3}{2 13 \cdot 2 13}

数字相乘:

7 \cdot 3 = 21

= \frac{21 3}{2 \cdot 2 13 13}

213 \cdot 213 = 52

213 \cdot 213

数字相乘:

2 \cdot 2 = 4

= 41313

使用根式运算法则:

a a = a

1313 = 13

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{21 3}{52}

\frac{3}{2 13} 1 - (3 \cdot \frac{3}{2 13})^{2} = \frac{5 3}{52}

\frac{3}{2 13} 1 - (3 \cdot \frac{3}{2 13})^{2}

1 - (3 \cdot \frac{3}{2 13})^{2} = \frac{5}{2 13}

1 - (3 \cdot \frac{3}{2 13})^{2}

(3 \cdot \frac{3}{2 13})^{2} = \frac{27}{52}

(3 \cdot \frac{3}{2 13})^{2}

乘

3 \cdot \frac{3}{2 13} : \frac{3 3}{2 13}

3 \cdot \frac{3}{2 13}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 3}{2 13}

= (\frac{3 \cdot 3}{2 13})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 3 ) ^{2}}{( 2 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(213)^{2} = 2^{2} (13)^{2}

= \frac{( 3 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(33)^{2} = 3^{2} (3)^{2}

= \frac{3 ^{2} ( 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

(13)^{2} : 13

使用根式运算法则:

a = a^{\frac{1}{2}}

= (1 3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 1 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 13

= \frac{( 3 ) ^{2} \cdot 3 ^{2}}{2 ^{2} \cdot 13}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3 \cdot 3 ^{2}}{2 ^{2} \cdot 13}

3 \cdot 3^{2} = 3^{3}

3 \cdot 3^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

3 \cdot 3^{2} = 3^{1 + 2}

= 3^{1 + 2}

数字相加:

1 + 2 = 3

= 3^{3}

= \frac{3 ^{3}}{2 ^{2} \cdot 13}

3^{3} = 27

= \frac{27}{2 ^{2} \cdot 13}

2^{2} \cdot 13 = 52

2^{2} \cdot 13

2^{2} = 4

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{27}{52}

= 1 - \frac{27}{52}

化简

1 - \frac{27}{52} : \frac{25}{52}

1 - \frac{27}{52}

将项转换为分式:

1 = \frac{1 \cdot 52}{52}

= \frac{1 \cdot 52}{52} - \frac{27}{52}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 52 - 27}{52}

1 \cdot 52 - 27 = 25

1 \cdot 52 - 27

数字相乘:

1 \cdot 52 = 52

= 52 - 27

数字相减:

52 - 27 = 25

= 25

= \frac{25}{52}

= \frac{25}{52}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{25}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

n ab = n a n b

= 13 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 213

= \frac{25}{2 13}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

= \frac{5}{2 13}

= \frac{5}{2 13} \cdot \frac{3}{2 13}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{3 \cdot 5}{2 13 \cdot 2 13}

213 \cdot 213 = 52

213 \cdot 213

数字相乘:

2 \cdot 2 = 4

= 41313

使用根式运算法则:

a a = a

1313 = 13

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{5 3}{52}

= \frac{21 3}{52} + \frac{5 3}{52}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{21 3 + 5 3}{52}

同类项相加:

213 + 53 = 263

= \frac{26 3}{52}

约分:

26

= \frac{3}{2}

\frac{3}{2} = \frac{3}{2}

真

代入

x = - \frac{3}{2 13} :

假

3 (- \frac{3}{2 13}) 1 - (- \frac{3}{2 13})^{2} + (- \frac{3}{2 13}) 1 - (3 (- \frac{3}{2 13}))^{2} = \frac{3}{2}

3 (- \frac{3}{2 13}) 1 - (- \frac{3}{2 13})^{2} + (- \frac{3}{2 13}) 1 - (3 (- \frac{3}{2 13}))^{2} = - \frac{3}{2}

3 (- \frac{3}{2 13}) 1 - (- \frac{3}{2 13})^{2} + (- \frac{3}{2 13}) 1 - (3 (- \frac{3}{2 13}))^{2}

去除括号:

(- a) = - a

= - 3 \cdot \frac{3}{2 13} 1 - (- \frac{3}{2 13})^{2} - \frac{3}{2 13} 1 - (- 3 \cdot \frac{3}{2 13})^{2}

3 \cdot \frac{3}{2 13} 1 - (- \frac{3}{2 13})^{2} = \frac{21 3}{52}

3 \cdot \frac{3}{2 13} 1 - (- \frac{3}{2 13})^{2}

1 - (- \frac{3}{2 13})^{2} = \frac{7}{2 13}

1 - (- \frac{3}{2 13})^{2}

(- \frac{3}{2 13})^{2} = \frac{3}{52}

(- \frac{3}{2 13})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 13})^{2} = (\frac{3}{2 13})^{2}

= (\frac{3}{2 13})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(213)^{2} = 2^{2} (13)^{2}

= \frac{( 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{2 ^{2} ( 13 ) ^{2}}

(13)^{2} : 13

使用根式运算法则:

a = a^{\frac{1}{2}}

= (1 3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 1 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 13

= \frac{3}{2 ^{2} \cdot 13}

2^{2} \cdot 13 = 52

2^{2} \cdot 13

2^{2} = 4

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{3}{52}

= 1 - \frac{3}{52}

化简

1 - \frac{3}{52} : \frac{49}{52}

1 - \frac{3}{52}

将项转换为分式:

1 = \frac{1 \cdot 52}{52}

= \frac{1 \cdot 52}{52} - \frac{3}{52}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 52 - 3}{52}

1 \cdot 52 - 3 = 49

1 \cdot 52 - 3

数字相乘:

1 \cdot 52 = 52

= 52 - 3

数字相减:

52 - 3 = 49

= 49

= \frac{49}{52}

= \frac{49}{52}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{49}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

n ab = n a n b

= 13 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 213

= \frac{49}{2 13}

49 = 7

49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

= \frac{7}{2 13}

= 3 \cdot \frac{7}{2 13} \cdot \frac{3}{2 13}

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 7 \cdot 3}{2 13 \cdot 2 13}

数字相乘:

7 \cdot 3 = 21

= \frac{21 3}{2 \cdot 2 13 13}

213 \cdot 213 = 52

213 \cdot 213

数字相乘:

2 \cdot 2 = 4

= 41313

使用根式运算法则:

a a = a

1313 = 13

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{21 3}{52}

\frac{3}{2 13} 1 - (- 3 \cdot \frac{3}{2 13})^{2} = \frac{5 3}{52}

\frac{3}{2 13} 1 - (- 3 \cdot \frac{3}{2 13})^{2}

1 - (- 3 \cdot \frac{3}{2 13})^{2} = \frac{5}{2 13}

1 - (- 3 \cdot \frac{3}{2 13})^{2}

(- 3 \cdot \frac{3}{2 13})^{2} = \frac{27}{52}

(- 3 \cdot \frac{3}{2 13})^{2}

乘

- 3 \cdot \frac{3}{2 13} : - \frac{3 3}{2 13}

- 3 \cdot \frac{3}{2 13}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{3 \cdot 3}{2 13}

= (- \frac{3 3}{2 13})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3 3}{2 13})^{2} = (\frac{3 \cdot 3}{2 13})^{2}

= (\frac{3 \cdot 3}{2 13})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 3 ) ^{2}}{( 2 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(213)^{2} = 2^{2} (13)^{2}

= \frac{( 3 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(33)^{2} = 3^{2} (3)^{2}

= \frac{3 ^{2} ( 3 ) ^{2}}{2 ^{2} ( 13 ) ^{2}}

(13)^{2} : 13

使用根式运算法则:

a = a^{\frac{1}{2}}

= (1 3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 1 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 13

= \frac{( 3 ) ^{2} \cdot 3 ^{2}}{2 ^{2} \cdot 13}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3 \cdot 3 ^{2}}{2 ^{2} \cdot 13}

3 \cdot 3^{2} = 3^{3}

3 \cdot 3^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

3 \cdot 3^{2} = 3^{1 + 2}

= 3^{1 + 2}

数字相加:

1 + 2 = 3

= 3^{3}

= \frac{3 ^{3}}{2 ^{2} \cdot 13}

3^{3} = 27

= \frac{27}{2 ^{2} \cdot 13}

2^{2} \cdot 13 = 52

2^{2} \cdot 13

2^{2} = 4

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{27}{52}

= 1 - \frac{27}{52}

化简

1 - \frac{27}{52} : \frac{25}{52}

1 - \frac{27}{52}

将项转换为分式:

1 = \frac{1 \cdot 52}{52}

= \frac{1 \cdot 52}{52} - \frac{27}{52}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 52 - 27}{52}

1 \cdot 52 - 27 = 25

1 \cdot 52 - 27

数字相乘:

1 \cdot 52 = 52

= 52 - 27

数字相减:

52 - 27 = 25

= 25

= \frac{25}{52}

= \frac{25}{52}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{25}{52}

52 = 213

52

52

质因数分解:

2^{2} \cdot 13

52

52

除以

252 = 26 \cdot 2

= 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

使用根式运算法则:

n ab = n a n b

= 13 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 213

= \frac{25}{2 13}

25 = 5

25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

= \frac{5}{2 13}

= \frac{5}{2 13} \cdot \frac{3}{2 13}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{3 \cdot 5}{2 13 \cdot 2 13}

213 \cdot 213 = 52

213 \cdot 213

数字相乘:

2 \cdot 2 = 4

= 41313

使用根式运算法则:

a a = a

1313 = 13

= 4 \cdot 13

数字相乘:

4 \cdot 13 = 52

= 52

= \frac{5 3}{52}

= - \frac{21 3}{52} - \frac{5 3}{52}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 21 3 - 5 3}{52}

同类项相加:

- 213 - 53 = - 263

= \frac{- 26 3}{52}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{26 3}{52}

约分:

26

= - \frac{3}{2}

- \frac{3}{2} = \frac{3}{2}

假

解是

x = \frac{3}{2 13}

x = \frac{3}{2 13}

将解代入原方程进行验证

将它们代入

arcsin (3 x) + arcsin (x) = 6 0^{\circ}

检验解是否符合

去除与方程不符的解。

检验

\frac{3}{2 13}

的解:

真

\frac{3}{2 13}

代入

n = 1

\frac{3}{2 13}

对于

arcsin (3 x) + arcsin (x) = 6 0^{\circ}

代入

x = \frac{3}{2 13}

arcsin (3 \cdot \frac{3}{2 13}) + arcsin (\frac{3}{2 13}) = 6 0^{\circ}

整理后得

1.04719 \dots = 1.04719 \dots

\Rightarrow 真

x = \frac{3}{2 13}

作图

流行的例子

4cos^2(2x)-4cos(2x)+1=0

4 cos^{2} (2 x) - 4 cos (2 x) + 1 = 0

cos(θ)= 2/(sqrt(13))

cos (θ) = \frac{2}{13}

4 sin (x - π) + 2 = 0

1/2 tan(2x)=tan(x)

\frac{1}{2} tan (2 x) = tan (x)

sin(x+pi)+cos(x+pi)=0

sin (x + π) + cos (x + π) = 0